2016/05/10 by Linhares, Andre, Swamy, Chaitanya · 1 citation
#Data Structures and Algorithms (cs.DS) #F.2.2 #FOS: Computer and information sciences #G.1.6 #G.2
paper · doi:10.48550/arxiv.1605.03203
We study the \em min-cost chain-constrained spanning-tree (abbreviated \mcst) problem: find a min-cost spanning tree in a graph subject to degree constraints on a nested family of node sets. We devise the \em first polytime algorithm that finds a spanning tree that (i) violates the degree constraints by at most a constant factor \em and (ii) whose cost is within a constant factor of the optimum. Previously, only an algorithm for \em unweighted \cst was known \citeolver, which satisfied (i) but did not yield any cost bounds. This also yields the first result that obtains an O(1)-factor for \em both the cost approximation and violation of degree constraints for any spanning-tree problem with general degree bounds on node sets, where an edge participates in a super-constant number of degree constraints. A notable feature of our algorithm is that we \em reduce \mcst to unweighted \cst (and then utilize \citeolver) via a novel application of \em Lagrangian duality to simplify the \em cost structure of the underlying problem and obtain a decomposition into certain uniform-cost subproblems. We show that this Lagrangian-relaxation based idea is in fact applicable more generally and, for any cost-minimization problem with packing side-constraints, yields a reduction from the weighted to the unweighted problem. We believe that this reduction is of independent interest. As another application of our technique, we consider the \em k-budgeted matroid basis problem, where we build upon a recent rounding algorithm of \citeBansalN16 to obtain an improved n^O(k1.5/ε)-time algorithm that returns a solution that satisfies (any) one of the budget constraints exactly and incurs a (1+ε)-violation of the other budget constraints.